Abstract:I prove three classification results about harmonic
morphisms
whose fibers have dimension one. All are valid when the
domain is at least of dimension 4. (The character of this
overdetermined problem is very different when the dimension
of the domain is 3 or less.)
The first result is a local classification for such
harmonic morphisms with specified target metric, the second
is a finiteness theorem for such harmonic morphisms with
specified domain metric, and the third is a complete
classification of such harmonic morphisms when the domain is
a space form of constant sectional curvature.
The methods used are exterior differential systems
and
the moving frame. The basic results are local, but,
because of the rigidity of the solutions, they allow a
complete global classification.